Lecture on Vectors

Introduction to Vectors

• Vector Quantities: Have both magnitude and direction.
  • Examples: Displacement, velocity, acceleration, force.
• Scalar Quantities: Have only magnitude, no direction.
  • Examples: Temperature, mass.

Distinction Between Scalars and Vectors

• Displacement vs Distance:
  • Displacement includes direction; distance does not.
  • Example: "45 meters east" is displacement; "45 meters" is distance.
• Velocity vs Speed:
  • Velocity includes direction; speed does not.
  • Speed tells how fast; velocity tells how fast and where.
• Acceleration:
  • A vector that indicates how fast velocity is changing.

Identifying Vectors and Scalars

• Force: Vector due to magnitude and direction (e.g., 100 N at 30° above x-axis).
• Mass: Scalar because it lacks direction (e.g., 10 kg).

Problem Example: Force Vector Components

• Given: Force vector of 100 N at 30° above the x-axis.
• Calculate:
  • X and Y components of the force vector.

Trigonometry and Vectors

• SOHCAHTOA: Used to solve vector components.
  • Sine (SO): ( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} )
    • ( f_y = f \times \sin(\theta) )
  • Cosine (CA): ( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} )
    • ( f_x = f \times \cos(\theta) )
  • Tangent (TO): ( \tan(\theta) = \frac{f_y}{f_x} )
    • Used to find angle using arc tangent.

Pythagorean Theorem and Vectors

• Formula: ( c^2 = a^2 + b^2 )
• For Vectors:
  • ( f = \sqrt{f_x^2 + f_y^2} )

Calculating Components

• X Component: ( f_x = 100 \times \cos(30°) )
  • Result: 86.6 N (using ( \cos(30°) = \sqrt{3}/2 )).
• Y Component: ( f_y = 100 \times \sin(30°) )
  • Result: 50 N (using ( \sin(30°) = 1/2 )).

Expressing Vectors

• Using Unit Vectors:
  • Unit Vector: Vector of magnitude one.
  • Standard Unit Vectors:
    • i: Represents x-axis.
    • j: Represents y-axis.
    • k: Represents z-axis.
• Expression: ( f = 86.6\ \text{i} + 50\ \text{j} )

Conclusion

• Vectors can be expressed in terms of magnitude and direction or as components using standard unit vectors.